Fountain pens use liquid ink. That ink can be purchased either in bottles or in sealed plastic cartridges.
Cartridges are installed by pressing them onto a short nipple at the back of the 'section' (the portion of the pen holding the nib). Cartridges are convenient because they are sealed and can easily travel. But they don't hold a lot of ink, and the selection of inks in cartridges is limited. And on a $/liquid ounce basis, ink in cartridges is more expensive than ink in bottles.
Most pen manufacturers offer refillable cartridges, aka converters. Some people also call these 'pumps'. I suspect the person who asked for a 'pump fountain pen' wants a pen with a converter. Converters also fit onto the nipple on the back of the section, but can be refilled. Some converters have a twist mechanism, others have a slider. Converters hold more ink and allow you to fill the pen from a bottle which gives you a broader choice of inks. Converters are also useful when the time comes to flush the pen because they can be used to force clean water through the section to remove any residual ink.
Some pen manufacturers use proprietary cartridges and converters, while others use a generic variety, sometimes referred to as 'standard international'. The components in pen kits all use standard international cartridges/converters, and every kit that I have purchased came with both a cartridge and a converter. You can purchase additional standard international converters from both CSUSA and PSI as well as other vendors.
There are a few other reservoir/filling mechanisms for fountain pens, but I'm not aware that these options are available in any kits. One option is the piston filler in which the body of the pen itself is the reservoir that is filled by twisting a knob hidden under a cap at the end of the barrel. Another option is eyedropper pens. These are pens in which the barrel is the reservoir that is filled using an eyedropper. Some fountain pen aficionados use standard cartridge/converter pens as eyedropper by removing the cartridge/converter, filling the body of the barrel with ink, and the using either a rubber o-ring or silicone grease to seal the joint between the section and barrel. You can usually tell those folks apart because of the fresh ink stains on their fingers and shirt pockets.